given some desired \(\varepsilon \) . The ranks found for \(\varepsilon \in [10^{-5},10^{-16}]\)  are shown in Figure REF .
The POD bases for \(\mathbf {A}_1\)  and \(\mathbf {A}_2\)  reach full rank \((r_1=15,\ r_2=45)\)  at \(\varepsilon = 10^{-6}\)  and \(\varepsilon = 10^{-8}\) , respectively.
Full-rank is not found for the basis of \(\mathbf {A}_3\ (r_3=240)\)  until \(\varepsilon = 10^{-16}\) .
This behavior is expected since compared to \(\mathbf {A}_3\) , the full ranks of \(\mathbf {A}_1\)  and \(\mathbf {A}_2\)  are relatively small. The singular values of both \(\mathbf {A}_1\)  and \(\mathbf {A}_2\)  also occupy a smaller range than for \(\mathbf {A}_3\) .
Another notable behavior is that \(r_3<r_2\)  for \(\varepsilon < 10^{-8}\) , indicating that the solution contained in the time range over which \(\mathbf {A}_2\)  was generated is the most difficult to represent with few POD modes.
This is to be expected given that \(\mathbf {A}_2\)  accounts for the solution during propagation of the radiation wave from the left boundary to the right, which is known to be a difficult phenomena for the POD to represent with low rank [1]}, [2]}. Let us note here that the rank of expansion for each timeframe in the F-C test is exactly the size of the linear system that solves for the coefficients \(\lambda _\ell \)  (Eq. (REF )). This means that when using \(\varepsilon = 10^{-5}\)  for instance, the largest linear system to solve in place of the BTE is a dense \(r \times r\)  system with \(r=14\) , which is of significantly lower dimensionality than the original BTE (\(r=14 \ll D=1.632\times 10^{4}\) ).
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